Abstract

Pixelated phase-mask interferograms have become an industry standard in spatial phase-shifting interferometry. These pixelated interferograms allow full wavefront encoding using a single interferogram. This allows the study of fast dynamic events in hostile mechanical environments. Recently an error-free demodulation method for ideal pixelated interferograms was proposed. However, non-ideal conditions in interferometry may arise due to non-linear response of the CCD camera, multiple light paths in the interferometer, etc. These conditions generate non-sinusoidal fringes containing harmonics which degrade the phase estimation. Here we show that two-dimensional Fourier demodulation of pixelated interferograms rejects most harmonics except the complex ones at {-3rd, +5th, −7th, +9th, −11th,…}. We propose temporal phase-shifting to remove these remaining harmonics. In particular, a 2-step phase-shifting algorithm is used to eliminate the −3rd and +5th complex harmonics, while a 3-step one is used to remove the −3rd, +5th, −7th and +9th complex harmonics.

© 2011 OSA

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References

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  3. O. Y. Kwon, “Multichannel phase-shifted interferometer,” Opt. Lett. 9(2), 59–61 (1984).
    [CrossRef] [PubMed]
  4. C. L. Koliopoulos, “Simultaneous phase-shift interferometer,” Proc. SPIE 1531, 119–127 (1992).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  7. M. Servin and J. C. Estrada, “Error-free demodulation of pixelated carrier frequency interferograms,” Opt. Express 18(17), 18492–18497 (2010).
    [CrossRef] [PubMed]
  8. B. Kimbrough and J. Millerd, “The spatial frequency response and resolution limitations of pixelated mask spatial carrier based phase shifting interferometry,” Proc. SPIE 7790, 1–12 (2010).
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]

2011 (1)

2010 (2)

M. Servin and J. C. Estrada, “Error-free demodulation of pixelated carrier frequency interferograms,” Opt. Express 18(17), 18492–18497 (2010).
[CrossRef] [PubMed]

B. Kimbrough and J. Millerd, “The spatial frequency response and resolution limitations of pixelated mask spatial carrier based phase shifting interferometry,” Proc. SPIE 7790, 1–12 (2010).

2009 (1)

2004 (1)

J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004).
[CrossRef]

2001 (1)

1996 (1)

1992 (1)

C. L. Koliopoulos, “Simultaneous phase-shift interferometer,” Proc. SPIE 1531, 119–127 (1992).
[CrossRef]

1984 (2)

R. Smythe and R. Moore, “Instantaneous phase measuring interferometry,” Opt. Eng. 23, 361–364 (1984).

O. Y. Kwon, “Multichannel phase-shifted interferometer,” Opt. Lett. 9(2), 59–61 (1984).
[CrossRef] [PubMed]

1982 (1)

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A 72(1), 156–160 (1982).
[CrossRef]

Brock, N.

J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004).
[CrossRef]

Estrada, J. C.

Gonzalez, A.

Hayes, J.

J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004).
[CrossRef]

Ina, H.

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A 72(1), 156–160 (1982).
[CrossRef]

Kimbrough, B.

B. Kimbrough and J. Millerd, “The spatial frequency response and resolution limitations of pixelated mask spatial carrier based phase shifting interferometry,” Proc. SPIE 7790, 1–12 (2010).

Kobayashi, S.

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A 72(1), 156–160 (1982).
[CrossRef]

Koliopoulos, C. L.

C. L. Koliopoulos, “Simultaneous phase-shift interferometer,” Proc. SPIE 1531, 119–127 (1992).
[CrossRef]

Kwon, O. Y.

Millerd, J.

B. Kimbrough and J. Millerd, “The spatial frequency response and resolution limitations of pixelated mask spatial carrier based phase shifting interferometry,” Proc. SPIE 7790, 1–12 (2010).

J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004).
[CrossRef]

Moore, R.

R. Smythe and R. Moore, “Instantaneous phase measuring interferometry,” Opt. Eng. 23, 361–364 (1984).

Ngoi, B. K. A.

North-Morris, M.

J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004).
[CrossRef]

Novak, M.

J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004).
[CrossRef]

Quiroga, J. A.

Servin, M.

Sivakumar, N. R.

Smythe, R.

R. Smythe and R. Moore, “Instantaneous phase measuring interferometry,” Opt. Eng. 23, 361–364 (1984).

Surrel, Y.

Takeda, M.

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A 72(1), 156–160 (1982).
[CrossRef]

Venkatakrishnan, K.

Wyant, J. C.

J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004).
[CrossRef]

Appl. Opt. (2)

J. Opt. Soc. Am. A (1)

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A 72(1), 156–160 (1982).
[CrossRef]

Opt. Eng. (1)

R. Smythe and R. Moore, “Instantaneous phase measuring interferometry,” Opt. Eng. 23, 361–364 (1984).

Opt. Express (3)

Opt. Lett. (1)

Proc. SPIE (3)

C. L. Koliopoulos, “Simultaneous phase-shift interferometer,” Proc. SPIE 1531, 119–127 (1992).
[CrossRef]

J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004).
[CrossRef]

B. Kimbrough and J. Millerd, “The spatial frequency response and resolution limitations of pixelated mask spatial carrier based phase shifting interferometry,” Proc. SPIE 7790, 1–12 (2010).

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Figures (6)

Fig. 1
Fig. 1

Periodic distribution for the basic building cell of the phase-mask p m ( x , y ) .

Fig. 2
Fig. 2

(Color online) (a) Non-sinusoidal fringe pattern having many harmonics overlapped at the spectral origin. (b) The same measuring phase but now modulated by the pixelated phase-mask over the CCD. As shown in Eqs. (6-9), some harmonics are spectrally displaced.

Fig. 3
Fig. 3

(Color online) (a) Multiplying the pixelated interferogram by the complex carrier leave the complex harmonics {-3rd, 5th, −7th, 9th,…} at the spectral origin distorting exp [ i ϕ ( x , y ) ] . (b) The phase estimation obtained by filtering-out outside the circle shown in (a), (Eq. (10)).

Fig. 4
Fig. 4

(Color online) Normalized spectral response of the 2-step (solid curve) and 3-step (dotted curve) temporal quadrature filters (Eqs. (14,15)). The complex harmonics marked with an “x” were suppressed through single-image spatial demodulation, while the vertical arrows represent those complex harmonics that survived this single-image demodulation process (Eq. (10)).

Fig. 5
Fig. 5

(Color online) Estimated phase and spectra obtained by spatio-temporal demodulation using (a) 2-step and (b) 3-step phase-shifted temporal samples (Eqs. (16,17)).

Fig. 6
Fig. 6

(Color online) Slices of the estimated phase using: (a) single-image pixelated interferogram demodulation (Eq. (10)), (b) 2-step PSA phase demodulation (Eq. (16)), and (c) 3-step PSA phase demodulation (Eq. (17)).

Equations (18)

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I ( x , y ) = a ( x , y ) + b ( x , y ) cos [ ϕ ( x , y ) + p m ( x , y ) ] ,
| p m ( x , y ) | > max | ϕ ( x , y ) | ,
I ( x , y ) exp [ i p m ( x , y ) ] = [ a + b cos ( ϕ + p m ) ] exp ( i p m ) , = a exp ( i p m ) + ( b / 2 ) { exp ( i ϕ ) + exp [ i ( ϕ + 2 p m ) ] } .
( 1 / 2 ) b ( x , y ) exp [ i ϕ ^ ( x , y ) ] = { exp [ i p m ( x , y ) ] I ( x , y ) } * h ( x , y ) .
tan [ ϕ ^ ( x , y ) ] = Im { [ exp ( i p m ( x , y ) ) I ( x , y ) ] * h ( x , y ) } Re { [ exp [ i p m ( x , y ) ] I ( x , y ) ] * h ( x , y ) } ,
I ( x , y ) = a ( x , y ) + n = 1 b n ( x , y ) cos { n [ ϕ ( x , y ) + p m ( x , y ) ] + θ n } ,
I ( x , y ) exp [ i p m ] = a exp ( i p m ) + n = 1 ( b n / 2 ) { exp [ i ( n 1 ) p m ] exp [ i ( n ϕ + θ n ) ] +                     exp [ i ( n + 1 ) p m ] exp [ i ( n ϕ + θ n ) ] } .
k = 1 , exp [ ± i ( 1 ) ( 0 π / 2 3 π / 2 π ) ] = exp [ ± i p m ( x , y ) ] , k = 2 , exp [ ± i ( 2 ) ( 0 π / 2 3 π / 2 π ) ] = exp [ ± i ( 0 π π 2 π ) ] = exp [ ± i π ( x + y ) ] , k = 3 , exp [ ± i ( 3 ) ( 0 π / 2 3 π / 2 π ) ] = exp [ ± i ( 0 3 π / 2 π / 2 π ) ] = exp [ ± i p m ( x , y ) T ] , k = 4 , exp [ ± i ( 4 ) ( 0 π / 2 3 π / 2 π ) ] = exp [ ± i ( 0 2 π 6 π 4 π ) ] = [ 1 1 1 1 ] ,
I ( x , y ) exp ( i p m ) = ( 1 / 2 ) { b 1 exp ( i ϕ ) + b 3 exp ( i 3 ϕ ) + b 5 exp ( i 5 ϕ ) + b 7 exp ( i 7 ϕ ) + } + ( 1 / 2 ) { b 1 exp ( i ϕ ) + b 3 exp ( i 3 ϕ ) + b 5 exp ( 5 i ϕ ) + } exp [ i π ( x + y ) ] + { a + b 2 cos ( 2 ϕ ) + b 4 cos ( 4 ϕ ) + b 6 cos ( 6 ϕ ) + b 8 cos ( 8 ϕ ) + } exp ( i p m ) .
A exp [ i ϕ ^ 1 ( x , y ) ] = [ exp ( i p m ) I ( x , y ) ] h ( x , y ) , = b 1 exp [ i ϕ ( x , y ) ] + n = 1 b 2 n + 1 exp [ i ( 1 ) n ( 2 n + 1 ) ϕ ( x , y ) ] ,
I ( x , y , t ) = a ( x , y ) + n = 1 b n ( x , y ) cos { n [ ϕ ( x , y ) + p m ( x , y ) + ω 0 t ] + θ n } , t = 0 , 1 , 2 , ...
exp [ i ϕ ^ 1 ]           = b 1 exp ( i ϕ ) + b 3 exp ( i 3 ϕ ) + b 5 exp ( i 5 ϕ ) + b 7 exp ( i 7 ϕ ) + exp [ i ( ϕ ^ 1 + ω 0 ) ]       = b 1 exp [ i ( ϕ + ω 0 ) ] + b 3 exp [ i 3 ( ϕ + ω 0 ) ] + b 5 exp [ i 5 ( ϕ + ω 0 ) ] + exp [ i ( ϕ ^ 1 + 2 ω 0 ) ] = b 1 exp [ i ( ϕ + 2 ω 0 ) ] + b 3 exp [ i 3 ( ϕ + 2 ω 0 ) ] + b 5 exp [ i 5 ( ϕ + 2 ω 0 ) ] +
s ( t ) = exp [ i ϕ ^ 1 ] δ ( t ) + exp [ i ( ϕ ^ 1 + ω 0 ) ] δ ( t 1 ) + exp [ i ( ϕ ^ 1 + 2 ω 0 ) ] δ ( t 2 ) .
          H 2 ( ω ) = 1 exp [ i ( ω + 3 ω 0 ) ] , h 2 ( t ) = F 1 [ H 2 ( ω ) ] = δ ( t ) exp ( i 3 ω 0 ) δ ( t + 1 ) ; ω 0 = π / 4.
H 3 ( ω ) = { 1 exp [ i ( ω + 3 ω 0 ) ] } { 1 exp [ i ( ω 5 ω 0 ) ] } , h 3 ( t ) = F 1 [ H 3 ( ω ) ] = δ ( t ) + [ 1 exp ( i ω 0 ) ] δ ( t + 1 ) + exp ( i 2 ω 0 ) δ ( t + 2 ) ; ω 0 = π / 3 ,
H 2 ( ω 0 ) exp ( i ϕ ^ 2 ) = { s ( t ) h 2 ( t ) } t = 2 = exp ( i ϕ ^ 1 ) e i 3 ω 0 exp [ i ( ϕ ^ 1 + ω 0 ) ] ; ω 0 = π / 4.
H 3 ( ω 0 ) exp ( i ϕ ^ 3 ) = { s ( t ) h 3 ( t ) } t = 3 = exp ( i ϕ ^ 1 ) + ( 1 e i ω 0 ) exp [ i ( ϕ ^ 1 + ω 0 ) ] + e i 2 ω 0 exp [ i ( ϕ ^ 1 + 2 ω 0 ) ] .
A exp [ i ϕ ^ N ( x , y ) ] = { exp [ i p m ( x , y ) ] [ I ( x , y , t ) * h N ( t ) ] t = N } * h ( x , y ) ,

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